Inﬂuence of nozzle random side loads on launch vehicle ... · Inﬂuence of nozzle random side...

Influence of nozzle random side loads on launch vehicle dynamicsNilabh Srivastava, Peter T. Tkacik, and Russell G. Keaninia�

Department of Mechanical Engineering, The University of North Carolina at Charlotte, Charlotte,North Carolina 28223, USA

�Received 8 March 2010; accepted 21 May 2010; published online 24 August 2010�

It is well known that the dynamic performance of a rocket or launch vehicle is enhanced when thelength of the divergent section of its nozzle is reduced or the nozzle exit area ratio is increased.However, there exists a significant performance trade-off in such rocket nozzle designs due to thepresence of random side loads under overexpanded nozzle operating conditions. Flow separationand the associated side-load phenomena have been extensively investigated over the past fivedecades; however, not much has been reported on the effect of side loads on the attitude dynamicsof rocket or launch vehicle. This paper presents a quantitative investigation on the influence ofin-nozzle random side loads on the attitude dynamics of a launch vehicle. The attitude dynamics oflaunch vehicle motion is captured using variable-mass control-volume formulation on a cylindricalrigid sounding rocket model. A novel physics-based stochastic model of nozzle side-load force isdeveloped and embedded in the rigid-body model of rocket. The mathematical model,computational scheme, and results corresponding to side loading scenario are subsequentlydiscussed. The results highlight the influence of in-nozzle random side loads on the roll, pitch, yaw,and translational dynamics of a rigid-body rocket model. © 2010 American Institute of Physics.�doi:10.1063/1.3457887�

I. INTRODUCTION

Given the economics of rocket launch and the need forhigher and more reliable dynamic performance, modernrockets are typically designed to achieve high thrust toweight ratios. This objective is often met using advancednozzle design modifications, including use of high nozzlearea ratios, reduced divergent section lengths, and optimizednozzle contours �e.g., TOC, TOP, CTP, etc.1�. Although suchnozzle designs theoretically predict higher vacuum perfor-mance, there is a significant performance trade-off associatedwith these rocket nozzles at sea-level operating conditions. Ithas been frequently reported1–3 that supersonic flow in suchnozzles tends to be overexpanded, which causes it to un-steadily detach/reattach itself to the nozzle wall. This conse-quently leads to generation of random nozzle side/lateralloads,4–7 which are often perilous in nature as they could notonly catastrophically affect the vibroacoustic response ofrocket, but also affect the safety margins associated with thetransient or first-stage of its attitude. In large engines, side-load magnitudes can be extremely large; for example, loadson the order of 250 000 pounds were typically observed dur-ing low altitude flight of the Apollo Program’s Saturn Vrockets.7 Indeed, minimizing and designing to accommodatepotentially catastrophic side loading represents an essential,long-standing design task within the rocket design commu-nity.

Supersonic flow separation in a rocket nozzle and theassociated side-load phenomena have been extensively in-vestigated both computationally and experimentally over thepast five decades. It is well-known1,8,9 that during overex-panded supersonic nozzle flow conditions, i.e., when Pe

�nozzle exit pressure� �Pa �altitude-dependent atmosphericpressure�, flow separation involving complex, three-dimensional, and mostly unsteady shock wave boundarylayer interactions �i.e., SWBLIs� occurs to compensate forthe adverse pressure gradient in the flow direction. This con-sequently generates large randomly fluctuating lateral forceson the nozzle structure. Turbulent SWBLI remains an exten-sively studied problem, both theoretically/computationallyand experimentally �refer to Chapman et al.,10 Zukoski,11

Dolling and Murphy,12 Sinha et al.,13 Polivanov,14 etc.�,where investigations where performed to study SWBLIs insupersonic flows under the presence of geometrical nonlin-earity or discontinuity �e.g., presence of a ramp, step, etc. inflow field�, capture unsteady interactions and its affect oninstantaneous pressure fluctuations, and predict/analyzeseparation zone geometry for various flow and wall coolingpatterns. With regard to flow separation and side loads inrocket nozzles, Östlund et al.1,4 provide comprehensive re-views of the state of the art, and emphasize the importance ofthree distinct side-load generation mechanisms: �a� randompressure variations due to free shock separation �FSS�, �b�shock transitions, i.e., FSS↔RSS �restricted shock separa-tion, where the separated boundary layer reattaches itselfdownstream of the separation point to form a recirculationzone�, and �c� aeroelasticity, which further amplifies the sideloads due to closed-loop fluid-structural vibroacoustic re-sponse. Original work on these features has been reported,e.g., by Onofri and Nasuti,15 Shimizu et al.,16 Frey andHagemann,17,18 Pekkari,19 and Schwane and Xia.20

Although work on rocket dynamics and control consti-tute vast, long studied areas of research �see, e.g., Refs.21–27�, the important question of attitude/ascent dynamicsof rockets subject to in-nozzle side loads, surprisingly, re-mains open. Likewise, vibration control and stability ofa�Electronic mail: [email protected].

JOURNAL OF APPLIED PHYSICS 108, 044911 �2010�

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http://dx.doi.org/10.1063/1.3457887

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launch vehicles subject to side loading remains poorlyserved. It is clear that the development of such control-oriented rocket dynamics models entails an embedding of afairly accurate description of side-load generation phenom-enon, which is a challenging task given the dependence ofside loads on a variety of inter-dependent interactions suchas internal and external flow patterns, nozzle geometry, am-bient conditions, fluid-structure interaction, etc. Although de-tailed computational fluid dynamics �CFD� modelingapproaches16,20 have been investigated to yield reasonablyaccurate predictions of separation point, wall pressure distri-bution, and nozzle side loads, such models, being computa-tionally intensive, render the synthesis of fast and reliablecontrollers difficult. Thus, significant research opportunitiesexist for development of accurate finite-dimensional system-level models suitable for studying the effects of random sideloads on rocket attitude dynamics, as well as investigation ofthe influence of side loads on the stability and elastic andrigid-body response of rockets as they move along theirlaunch trajectory.

The present study has two objectives. First, a simple, yetrealistic model of rocket dynamics is sought which takes intoaccount the effects of stochastic, altitude-dependent, in-nozzle side loads. The model will be used to study the influ-ence of these loads on rocket center-of-mass dynamics, sta-bility, and flight trajectory. Given the simplicity of the model,it is anticipated that it could provide a basis for developmentof, e.g., robust trajectory control strategies and/or structuralvibration suppression techniques, particularly during low al-titude flight when side loading can be significant.

The second objective of this work centers on develop-ment of a simple, physically realistic model of random side-load generation �especially under FSS regime� and evolutionduring over-expanded low altitude flight. It is clear thatknowledge of reasonably accurate separation location andseparation criteria will facilitate the development of thesephysics-based side-load prediction models. A number ofcriteria1,3,28,29 have been proposed in past for predicting thenominal FSS point. The side-load prediction model presentedin this work is an extended modification of Keanini andBrown,29 where simple, yet physics-based, scale analyses oftransverse momentum transport across the separating bound-ary layer have been used to derive separation criteria fortime-average turbulent SWBL pressure fluctuations in over-expanded nozzles operating under FSS regime. The proposedmodel focuses on the random shape and motion of the in-stantaneous in-nozzle boundary layer separation line, and incontrast to existing statistically-based models, requires rela-tively little experimental or numerical data on the separation-zone wall pressure distribution.

In brief overview, the paper first presents the rocket dy-namics model. The proposed model uses a control volumeapproach, accounts for six degrees of coupled rigid bodytranslational and rotational motion, incorporates a simple,altitude-dependent model of external aerodynamic loading,and accounts for in-nozzle side loads. The model for random,altitude-dependent side loads is then described. Here, a scal-ing argument is presented indicating that random, spatiallyvarying nozzle-wall pressure distributions immediately up-

and down-stream of the instantaneous, azimuthally-varyingboundary layer separation line are small relative to the com-paratively large, altitude-dependent, mean downstream pres-sure. This result in turn allows straightforward calculation ofthe instantaneous side load, given an instantaneous realiza-tion of the random separation line shape. It is shown that themodel provides straightforward, analytical explanations forseveral well known side-load statistical properties.

The rocket dynamics model is then used to study thestochastic response of sounding-rocket-scale launch vehiclessubject to low altitude random side loads. Although sideloads appear only during the earliest portion of flight, theirinfluence on subsequent evolution of pitch, yaw, and lateraldisplacement is significant; individual realizations of randomrocket motion are thus described, as well as ensemble aver-aged translational and rotational rocket dynamics. Finally,the paper closes with suggestions for future work.

II. ATTITUDE DYNAMICS MODEL OF A RIGID ROCKET

Since the primary objective of this research is to studythe influence of nozzle side loads on rocket launch dynamics,a canonical rigid-body rocket model is considered, where thevariable-mass and flow dynamics are captured in a compre-hensive manner using control volume formulation. Figure 1illustrates the geometrical description of rocket model alongwith the forces acting on it. The rocket is subjected to adeterministic, time-dependent aerodynamic load, a time-varying deterministic thrust load, and a stochastic in-nozzleside load. The following assumptions are made during thecourse of model development.

• Rocket body is axisymmetric at all times.• The internal flow of burnt products is axisymmetric

and steady.• The instantaneous mass center lies on the longitudinal

axis �i.e., the axis of symmetry� at all times and doesnot undergo significant variation from its initial con-figuration.

• The line of action of aerodynamic load is along thelongitudinal axis of symmetry, i.e., ignore the effectsof variable angle of attack �since the primary focus isto investigate the effects of nozzle side loads on rocketdynamics�.

• Neglect the effects of stochastic wind loads on rocketdynamics.

• The exhaust gas flow is axisymmetric, uniform, andsteady.

Using Reynolds transport theorem and Newton–Euler’s mo-mentum equation for a control volume accelerating in a non-inertial frame of reference, one gets

F� S + F� B =�

�t�

CVv��dV + �

CSv��v� . dA� �

+ �M

�a�o + 2�� v�� + �� r� + �� r��dM .

�1�

Here, F� S and F� B denote all the surface and body forces acting

044911-2 Srivastava, Tkacik, and Keanini J. Appl. Phys. 108, 044911 �2010�


on the control volume of the rocket. �, v� , and r�, respectively,denote the instantaneous density, velocity, and position fields�relative to the control volume� for the flow of combustionproducts. r� specifically denotes the distance of an infinitesi-mal fluid particle from the instantaneous center of mass O,whereas, �� denotes the instantaneous angular velocity of thebody-fixed coordinate axes attached to the rocket at its centerof mass O. a�o denotes the acceleration of the center of massO of the rocket with respect to an inertial frame of reference�i.e., XYZ in Fig. 1�. Given O as the instantaneous center ofmass of rocket �i.e., �Mr�dM =0� where the body-fixed refer-ence frame is attached and the assumption of steady internalflow of burnt products �i.e., �� • � /�t�0�, Eq. �1� could befurther simplified as

F� S + F� B − �CS

v��v� . dA� � − �M

2�� v��dM = Ma�o. �2�

The surface and body forces acting on the rocket can bereadily expressed as

F� S = F� A + �pe − pa�Aei� + Fsyj� + Fszk� ,

F� B = − MgI�. �3�

Here �I� ,J� ,K� � represent the unit vectors of the inertial coor-dinate axes ”XYZ,” while �i�, j� ,k�� represent unit vectors ofthe body-fixed coordinate axes “xyz” with origin at O �therocket’s instantaneous center of mass�, as depicted in Fig. 1.Also, pe is the gas pressure at nozzle exit plane and pa is theambient pressure. Recognizing the need to emphasize thedynamic influence of in-nozzle stochastic side loads Fsy andFsz, we seek a simple though qualitatively reasonable modelof the instantaneous aerodynamic load �F� A� as

F� A = −1

2CD�a�xo

2 + yo2 + zo

2�ARi�. �4�

Thus, the aerodynamic load on the rocket is primarily ap-proximated as a drag force with its line of action coincidingwith the longitudinal axis of symmetry �or the x-axis of thebody-fixed coordinate frame�. Although precise or detailedeffects of the angle of attack are not captured by the above-mentioned aerodynamic load model, the Mach number-dependent drag-coefficient CD �refer to Fig. 2 �Ref. 30�� andthe instantaneous changes in the center of mass velocity mayserve as useful indicators for analyzing the effects corre-sponding to variable angle of attack.

Using the mass conservation principle, it is easy to ob-tain the rate at which mass is being depleted from the rocketcontrol volume

dM

dt= − �

CS�v� . dA� � . �5�

Since the rocket loses mass only through the nozzle exit area�Ae�, Eq. �5� could be written as

dM

dt= − �

Ae

��v� . dA� � . �6�

Using the assumption of steady axisymmetric exhaust flowand the axisymmetry of rocket nozzle, Eq. �6� yields an ex-pression for the rate of mass loss from the rocket

M = − �evexAe. �7�

The above expression for mass loss rate encapsulates theassumption that the density of exhaust gas ��e� does notchange appreciably over the nozzle exit plane. vex is themagnitude of exhaust gas velocity relative to rocket body

FIG. 1. �Color online� Geometrical description of rigid-body rocket model.



computed at the nozzle exit plane along the x-direction �i.e.,the longitudinal axis of the rocket� of the body-fixed refer-

ence frame. Note M is constant if the flow through nozzlethroat is choked. Using Eqs. �3�, �4�, and �7�, Eq. �2� couldbe modified to get

−1

2CD�a�xo

2 + yo2 + zo

2�ARi� + �pe − pa�Aei� + Fsyj� + Fszk� − MgI�

+ Mvexi� − 2�� M

v�dM = M�xoI� + yoJ� + zoK� � . �8�

Using Reynolds transport theorem, the Coriolis term in Eq.�8� could be expanded to obtain

−1

2CD�a�xo

2 + yo2 + zo

2�ARi� + �pe − pa�Aei� + Fsyj� + Fszk� − MgI�

+ Mvexi� − 2��

�t�

CV�r�dV + �

CS�r��v� . dA� ��

= M�xoI� + yoJ� + zoK� � . �9�

Again, under the assumption of steady internal flow and neg-ligible variations in center of mass from its initial configura-tion, Eq. �9� reduces to the following form:

−1

2CD�a�xo

2 + yo2 + zo

2�ARi� + �pe − pa�Aei� + Fsyj� + Fszk�

− MgI� + Mvexi� − 2�� − M�L − b�i�� = M�xoI�

+ yoJ� + zoK� � . �10�

Expressing the angular velocity �� in body-fixed coordinateframe as �� =�xi�+�yj�+�zk� and introducing the Eulerian roll��—pitch ��—yaw �� transformation from �i�, j� ,k�� refer-ence frame to the inertial �I� ,J� ,K� � frame

� i�

j�

k� = � cos � sin � 0

− sin � cos � 0

0 0 1��cos � 0 − sin �

0 1 0

sin � 0 cos ��

��1 0 0

0 cos � sin �

0 − sin � cos �� I�

J�

K� , �11�

the following three scalar governing equations for center-of-mass dynamics of rocket could be obtained:

X-motion

Mxo = ��pe − pa�Ae + Mvex − 0.5CDAR�a�xo2 + yo

2

+ zo2��cos � cos � − Mg − �Fsy + 2M�L

− b��z�cos � sin � + �Fsz − 2M�L − b��y�sin � , �12�

Y-motion

Myo = ��pe − pa�Ae + Mvex − 0.5CDAR�a�xo2 + yo

2 + zo2��

��cos � sin � + sin � sin � cos �� + �Fsy + 2M

��L − b��z��− sin � sin � sin � + cos � cos ��

− �Fsz − 2M�L − b��y�sin � cos � , �13�

Z-motion

Mzo = ��pe − pa�Ae + Mvex − 0.5CDAR�a�xo2 + yo

2 + zo2��

��sin � sin � − cos � sin � cos �� + �Fsy + 2M

��L − b��z��cos � sin � sin � + sin � cos ��

+ �Fsz − 2M�L − b��y�cos � cos � . �14�

It is evident from Eqs. �12�–�14� that in-nozzle stochas-tic side loads could significantly influence the translationaldynamics of a rocket during its attitude. Also, it is to benoted that the Euler angles �� ,� ,�� in these equations time-

FIG. 2. �Color online� A typical representation of dragcoefficient �CD� vs Mach number for a rocket �Ref. 30�.



varyingly depend on the angular velocity �� , which furtherhighlights the coupling between the translational and rota-tional �i.e., roll, pitch, yaw� motions of rocket during its at-titude. Using the rotation matrices, one could readily obtainthe following relationship between the components of angu-lar velocity in body-fixed reference frame and the time rateof change in Euler angles

��x

�y

�z = � cos � sin � 0


0 0 1��cos � 0 − sin �

0 1 0

sin � 0 cos ��

0

0

+ � cos � sin � 0


0 0 1��0

�

0 + �0

0

�

or ��x

�y

�z = � cos � cos � sin � 0

− cos � sin � cos � 0

sin � 0 1��

�

� . �15�

It is clear from Eq. �15� that a unique correspondence existsbetween the time rate of change in Euler angles and thecomponents of angular velocity as long as the above trans-formation matrix is nonsingular �i.e., unless �= /2�. Forthe instant when �= /2, the time rates of Euler angles areobtained using the following equations:

� = ��x2 + �y

2,

� =�y�x − �x�y

�x2 + �y

2 ,

� = ��z − ��sgn�sin ��, � � �−

2,

2� . �16�

Using generalized Kane’s equations, Eke et al.27,31 de-rived the following vector equation for the rigid-body rota-tional dynamics of a variable mass system.

I�� + �� I�� +dI�

dt�� + �

CS��r� � �� r��v� . dA� �

+�

�t�

CV��r� � v��dV + �

CV�� r� � v��dV

+ �CS

��r� � v��v� . dA� � = M� ext. �17�

Here, I� denotes the principal inertia tensor about the body-fixed axes �xyz�, which are also the principal axes. Using theaxisymmetry of rocket body, it is assumed Iyy = Izz= I. Thecomputation of the surface and volume integral terms in Eq.�17� could be readily done as explained subsequently.

The position vector �as shown in Fig. 1� from the instan-taneous mass center O to a generic fluid particle leavingnozzle exit area �Ae� could be expressed as

r�Ae= − �L − b�i� + r cos �j� + r sin �k� .

The infinitesimal fluid area element at nozzle exit could beexpressed as dA� =−rdrd�i�. The exhaust gas velocity profile�relative to rocket body� over the nozzle exit plane �Ae� isconsidered to be31

v� Ae= − vexi� + ��xr

2

Re− �xr��− j� sin � + k� cos �� . �18�

The j� and k� components in the above expression �similar tothose introduced by Tran and Eke31� account for the effectsdue to whirling of fluid particles as the rocket �primarilyassumed to be a cylindrical body� spins or rolls about itslongitudinal axis �i.e., the x-axis of the body-fixed referenceframe�. The term �xr

2 /Re assumes parabolic distribution ofazimuthal velocity field as the fluid particles escape/cross thenozzle exit plane. This is a simplified approximation thataccounts for the fact that the fluid particles on the longitudi-nal axis of symmetry �i.e., x-axis� do not whirl, while thoseparticles at the nozzle surface will whirl with a tangential/peripheral azimuthal velocity of �xRe. Note vex is not a func-tion of �, owing to axisymmetry, and is also assumed to beuniform/constant over the nozzle exit plane. Also, density �;is assumed to be uniform over the nozzle exit plane. Usingthese expressions, the surface integral terms in Eq. �17�could be readily evaluated to yield the following expres-sions:

�CS=Ae

��r� � v��v� . dA� � = −M10

�xRe2i�. �19�

�CS=Ae

��r� � �� r��v� . dA� � = M��L − b�2 + 0.25Re2�

��yj� + �zk�� + 0.5Re2�xi�� . �20�

For the volume integral term �i.e., the sixth term in Eq.�17��, the contributions from geometrical nonuniformities inrocket body and nozzle sections are neglected and the rocketcombustion chamber is primarily treated as a cylinder of ra-dius Ri with an axisymmetric internal flow field having avelocity profile �relative to rocket body� similar to Eq. �18��Ref. 31�

v� = − vxi� + � r

Ri− 1��xr�− j� sin � + k� cos ��; vx � f�� .

�21�

The position vector from the instantaneous mass center O toa generic fluid particle contained within the rocket body isgiven by r�= xi�+r cos �j�+r sin �k�. The infinitesimal fluid vol-ume element could be expressed as dV=rdrd�dx. Usingthese expressions, the volume integral term �i.e., the sixthterm in Eq. �17�� could be evaluated to yield the followingexpression:



�CV

�� r� � v��dV = −MLRi

4

10vexRe2 ��x�zj� − �x�yk�� .

�22�

It is clear that this volume integral term in fact captures thegyroscopic torques that the rocket experiences during its at-titude. Using Eqs. �19�, �20�, and �22� and the assumption ofsteady internal flow, Eq. �17� could be simplified to yield thefollowing three governing equations for the rotational dy-namics of rocket during its attitude:

Ixx�x + �Ixx +2

5MRe

2��x = 0, �23�

I�y + �Ixx − I��x�z + I�y + M��L − b�2 + 0.25Re2��y

−MLRi

4

10vexRe2�x�z = M� ext · j�, �24�

I�z − �Ixx − I��x�y + I�z + M��L − b�2 + 0.25Re2��z

+MLRi

4

10vexRe2�x�y = M� ext · k� . �25�

Since the line of action of aerodynamic load is assumed tocoincide with the longitudinal axis �i.e., x-axis of body-fixed

frame� through the center of mass O of the rocket, the onlyforce that contributes to the external moment term M� ext inEqs. �24� and �25� is the in-nozzle stochastic side load. Themoment due to these stochastic side loads could be computedas

M� ext = − �L − b�i� � �Fsyj� + Fszk�� . �26�

Thus, Eqs. �12�–�16� and �23�–�26� together constitute thegoverning equations for translational and rotational dynam-ics of a variable-mass rigid-rocket during its attitude. It isclear from these equations that the stochastic lateral loads onthe nozzle walls could significantly influence the dynamicresponse of rocket, which could be undesirable or perilous.The subsequent section discusses the mathematical model forcomputation of these stochastic side loads �i.e., Fsy and Fsz�on the rocket nozzle.

It is evident from the governing equations of rocketmodel that a fairly accurate description of altitude-dependentatmospheric pressure and density is needed for capturing theoverexpanded flow dynamics. Thus, a National Aeronauticsand Space Administration �NASA� atmospheric model isadopted from literature32 and embedded in the above govern-ing equations. The NASA atmospheric model for ambientpressure, temperature, and density is given by the followingequations:

pa�in kPa� =�101.29�Ta + 273.1

288.08�5.256

, xo � 11000 m�i.e. meters�

22.56e�1.73−0.000157xo�, 11000 m � xo � 25 000 m,

2.488�Ta + 273.1

216.6�−11.388

, xo � 25 000 m Ta�in ° C� = �15.04 − 0.006 49xo, xo � 11 000 m

− 56.46, 11 000 m � xo � 25 000 m,

− 131.21 + 0.002 99xo, xo � 25 000 m

�a�in kg/m3� =pa

0.2869�Ta + 273.1�.

III. NOZZLE SIDE-LOAD MODEL FOR FSS REGIME

Broadly speaking, the origin of side loads on rocketnozzles can be traced to one or more of the following fea-tures: steady and/or time varying asymmetries in the nozzleflow and pressure fields, steady and/or time-varying asym-metries in nozzle shape, and external aerodynamic loading.Focusing on the first case, the following chain of physicalprocesses can create side loads.

A. Side load physical processes

During low altitude flight, ambient pressure can, and of-ten does, exceed near-exit pressures within the nozzle, i.e.,the nozzle flow can be overexpanded. Under these condi-tions, excess external pressure can force ambient air up-stream into the nozzle, where the incoming flow is confinedto the low inertia near-wall region. This counter-flow contin-ues upstream to a locus of points, the nominal boundarylayer separation line, s� , t� �as depicted in Fig. 3�, at whicha balance between �decaying� upstream inertia and down-stream boundary layer inertia causes separation of the latter.



The separating boundary layer, in turn, forms a virtual turn-ing corner along the nozzle wall, triggering formation of athree-dimensional oblique shock structure within the externalsupersonic, nonboundary layer flow. The shape of s� , t� israndom and asymmetric. Thus, due to the pressure jumpacross the associated oblique shock, a net, nominally-radialpressure force, the side load, Fr�t� (boldface font representsvector quantity), acts on the nozzle wall. Due to the random,time-varying shape of s� , t�, the instantaneous magnitude,A�t�= Fr�t�=�Fsy

2 +Fsz2 , and direction, , of Fr likewise vary

randomly in time. Two distinct shock-boundary layer sepa-ration structures, the FSS and RSS structure, appear to playprominent roles in nozzle side loading.1,4–6 Although themodel proposed here applies to side loads associated with theFSS structure, a similar approach can be adapted to sideloading associated with RSS structures.

Current side-load models can be characterized as one oftwo types, phenomenological models which attribute sideloading to, e.g., a fixed boundary layer separation line withinthe nozzle,33 or, more recently, semiempirical statisticalmodels4,34,35 which require experimentally measured correla-tions of the nozzle wall pressure field. Dumnov34 introducedthe latter approach in 1996 and his ideas now dominate thisarea of research.

The objectives of this section are threefold. First, wewish to propose an alternative probabilistic approach toDumnov for computing side loads. As described here, thepresent approach focuses on the statistical behavior of therandom separation line, s� , t�. Second, closure of existingprobabilistic side-load models requires either complex ex-perimental measurements of nozzle wall pressure distribu-tions, or development of high-level compressible flow simu-lations capable of capturing complex three-dimensional,unsteady shock boundary layer interactions.1,35,36 We wish todevelop a relatively simple, physically consistent model ofseparation line motion and side loading that circumvents the

heavy experimental and numerical modeling demands asso-ciated with present approaches. Specifically, we propose apurely analytical solution to the closure problem. Third,while altitude effects play a crucial role in side-load evolu-tion and behavior, this important feature has not been exam-ined. Thus, we incorporate this effect within the proposedmodel.

The proposed side-load model requires statistical infor-mation on two random features: �i� the instantaneous azi-muthal pressure distribution in the vicinity of the instanta-neous in-nozzle boundary layer separation line, s� , t�, and�ii� the instantaneous, azimuthally varying shape of s� , t��refer to Figs. 3 and 4�.

With regard to the first feature, scaling arguments belowindicate that at any instant, spatial pressure fluctuations im-mediately upstream and downstream of the instantaneousseparation line are small relative to associated �spatially uni-form� mean pressures. At first glance, this appears to contra-dict the well known observation34 that pressures near theseparation line exhibit significant random variations in boththe axial and azimuthal directions. However, based on ourscaling analysis, we argue that these observations reflect ran-dom fluctuations in the separation line shape, taking placewithin near-uniform upstream and downstream wall pressurefields.

Comparing experimental requirements necessary for clo-sure of Dumnov’s model34 versus those required for closureof the present model, since Dumnov34 ignores separation linedynamics, his approach again requires experimentally or nu-merically generated data on the axially and azimuthally vary-ing nozzle wall pressure distribution �obtained in the vicinityof the boundary layer separation zone�. The present ap-proach, by contrast, exploits the well-known observation37–40

that local separation line dynamics exhibit fairly universalstatistical characteristics, independent of the shock generator,nozzle type, and separation location. �Here, and with refer-

FIG. 3. Schematic of instantaneous boundary layer separation line shape,s� , t�. The time, or equivalently, altitude-dependent mean separation linelocation �along the nozzle’s longitudinal axis� is denoted as xs�H�t��, whereH�t� is the instantaneous rocket altitude. The corresponding nozzle radius isR�xs�.

Δφ

�

�

xx

s

s(φ

φ i

i,t)

Ls

φ

(H(t

))

R(H

(t))

FIG. 4. Separation line model: the mean separation line position, xs�H�t��,moves down the nozzle axis, x on the slow time scale associated withvertical rocket motion. By contrast, axial separation line motion aboutxs�H�t��, at any angular position, i, is random, and takes place on a muchshorter time scale; rapid axial motion, in addition, is confined to the nominalshock boundary layer interaction zone, denoted by Ls. Pressures upstream�Pi� and downstream �P2� of the instantaneous separation line, s� , t�, areassumed to be spatially uniform within Ls, but do vary with rocket altitude,H�t�.



ence to Fig. 4, local refers to separation line motion observedwithin a thin rectangular region of �say� lateral width R� and axial length Ls, where R and Ls are defined below and inthe caption.� Presuming that the statistics of separation linemotion remain nominally independent of azimuthal position�within circular nozzles�, the experimental effort required forclosure here thus appears to be significantly less; again, weuse simple analytical modeling in order to achieve closure.

B. Probabilistic side-load model

Considering the instantaneous force vector produced byasymmetric boundary layer separation, Fs�t�, expressed as asum of radial and axial components

Fs�t� = Fr�t� + Fx�t� , �27�

we note the following important experimental and numericalobservations concerning the side load, Fr, �within rigid, axi-symmetric nozzles�:

�a� the probability density of the random amplitude, A= Fr, is a Rayleigh distribution34,35 and

�b� the random instantaneous direction, , of Fr is uni-formly distributed over the periphery of the nozzle, orp � �=1 /2, where p is the pdf of the side-loaddirection.35

In this subsection, we adapt a discussion from Ref. 41 toshow that both observations can be explained using a simplestatistical model of random side loads. Knowing A and , itis trivial to express the instantaneous side-load componentsin body-fixed y- and z-directions as Fsy =A cos and Fsz

=A sin . The following assumptions were made regardingthe statistics of Fsy and Fsz:

�i� Fsy and Fsz are independent, Gaussian random vari-ables,

�ii� �Fsy�=0 and �Fsz�=0, and�iii� ��Fsy − �Fsy��2�= ��Fsz− �Fsz��2�=�2,

where, assuming ergodicity, � · � denotes either an ensembleor time average. As shown in subsection D, the first twoassumptions can be derived from the model of separationline dynamics presented there and in subsection C, the lastassumption reflects the presumption that the random flowfeatures underlying side loading are azimuthally homoge-neous.

Thus, writing Fsy and Fsz as, Fsy = Y =A cos and Fsz

= Z=A sin , the joint probability density associated with Fsy

and Fsz can be expressed as

pYZ�Y,Z� = pY�Y� · pZ�Z� =1

2�2exp�−Y2 + Z2

2�2 � . �28�

Following Ref. 41, we restate pYZ in terms of A and as,

pA = JpYZ�Y,Z� , �29�

where pA �A , � is the joint pdf for the random amplitudeand direction of Fr, and where the Jacobian determinant isgiven by

J = ��Y

�A

�Y

�

�Z

�A

�Z

�

= A� . �30�

Thus,

pA �A, � =A

2�2exp�−A2

2�2� = � 1

2� A

�2exp�−A2

2�2��= p � �pA�A� , �31�

where

p � � =1

2, 0 � � 2 , �32�

is again the uniform probability density underlying the ran-dom direction and

pA�A� =A

�2exp�−A2

2�2� , �33�

is the Rayleigh distribution for the amplitude A.It is thus clear that the simple assumptions �i�–�iii� above

provide a basis for explaining and modeling known side-loadstatistical properties. In addition, this appears to be the firstanalytical, i.e., nonexperimental and nonnumerical, explana-tion of the observations34,35 noted in �a� and �b� above.

C. Model closure: Separation line shape

In order to close the statistical description of randomside loads, the parameter � in Eqs. �31� or �33� must bedetermined. In this section, we

�i� relate � to �A2� and�ii� propose a model of separation line dynamics.

The second task rests on simple scale analyses of thefluid dynamical features extant within, and near, the shock-boundary layer interaction zone, as well as introduction ofsimple assumptions on the statistics of separation line mo-tion. Given the separation line model, the side-load modelcan then be closed, as described in subsection D.

The parameter � can be related to �A2� by first notingfrom the statistical model of the side-load components Fsy

and Fsz, that

��Fsy − �Fsy��2� = �Fsy2 � = �2,

��Fsz − �Fsz��2� = �Fsz2 � = �2. �34�

Since, A2=Fsy2 +Fsz

2 , we get

�A2� = �Fsy2 � + �Fsz

2 � = �2 + �2 = 2�2 �35�

or

� =1�2

��A2� . �36�

Equation �36� could also have been directly obtained usingstandard formulae for the Rayleigh distribution41 �A�=�� /2, var�A�=�2�4−� /2, and var�A�= �A2�− �A�2.



1. Model of separation line motion

Considering the random axial �streamwise� motion ofthe boundary layer separation line, we take advantage of aseparation in time scales, �R and �s, associated, respectively,with the slow downstream motion of the line’s mean posi-tion, xs�t�=xs�xo�t�=H�t��, and the rapid motion of the sepa-ration line about xs�t�. The mean position of the separationline moves downstream in response to the decaying externalambient pressure; thus, �R is estimated as �R=�Ha /VR,where �Ha is the characteristic incremental altitude overwhich significant ambient pressure changes occur and VR is acharacteristic rocket speed. By contrast, �s corresponds to thelower end of the frequency spectrum associated with largeamplitude, random axial motion of the separation line aboutxs; this lower end ranges from approximately 10 to 300 Hzwhile the amplitude of random axial motion, delimiting thenominal shock-boundary layer interaction zone, ranges fromapproximately 1 to 5 cm.39

Thus, since �R��s, then over time intervals �t=O��R�,well defined statistical features associated with the fast sepa-ration line motion about xs�t� can, at any given instant, bereliably determined. Given this difference in time scales, wepropose the following model of separation line motion:

�i� Assume that at any altitude H=H�t�=xo�t�, a station-ary, time �or equivalently, ensemble� average separa-tion line shape, s�H�, exists, where averaging is car-ried out over intervals T that are long relative to �s,but short relative to �R.

�ii� Assume that the mean separation line shape, s�H�, isindependent of the azimuthal angle . This is a rea-sonable assumption for FSS within nominally sym-metric nozzles that are attached to well-designed com-bustion chambers that do not produce significantasymmetric combustion.

�iii� At any altitude H, or equivalently, any time t, dis-cretize the instantaneous separation line shape into Nequiangular increments, � . As shown in Fig. 4, wedefine a circular reference line passing around the in-ner periphery of the nozzle, where the reference linecoincides with the mean axial separation line location,xs�H�t��. In addition, define N differential areas

�Ai = R�xs�t��s� i,t�� , i = 1,2, . . . ,N , �37�

where R�xs�t�� is the nozzle inner radius at xs�t�=xs�H�t��, and s� i , t� is the instantaneous axial posi-tion of the separation line at = i, relative to the�time-varying� reference line.

�iv� Define, at any given altitude H, a shock-boundarylayer interaction zone of axial length Ls which encom-passes the axial region over which the separation linemoves. Assume that within this zone pressures up-stream and downstream of the instantaneous separa-tion line, Pi�t�= Pi�xs�H�t�� and P2�t�= P2�H�t��, re-spectively, are independent of and only depend onaltitude H=H�t�.

In order to justify these assumptions, and as animportant aside prior to listing the last two model as-sumptions, we use scaling to argue that spatial pres-

sure variations both up- and downstream of xs�t�,within the nominal shock-boundary layer interactionzone, are small relative to the respective �background,slowly time varying� mean pressures, Pi�t� and P2�t�.

Considering first the upstream side of the instan-taneous separation line, three potential sources of spa-tial pressure variations can be identified: azimuthalacoustic pressure modes within the upstream inviscidsupersonic flow, upstream transmission of acousticdisturbances within subsonic portions of the turbulentboundary layer, and dynamic pressure produced bythe turbulent boundary layer. Pressure variations pro-duced by azimuthal acoustic modes are likely minimalsince these modes cannot propagate �azimuthally�more than a distance of order O�Ls /Mi� �where Mi isthe free stream Mach number at the incipient separa-tion point�.

With regard to the second source, while Liep-mann et al.42 observed that acoustic disturbancestravel no more than one or two boundary layer thick-nesses upstream within turbulent compressible bound-ary layers, in relatively thick boundary layers, suchdisturbances could produce spatial pressure variationson the upstream side of the instantaneous separationline. However, extending an argument given immedi-ately below, since the maximum characteristic magni-tude of these variations, �p�, is small relative to thecharacteristic pressure difference, P2− Pi across theseparation line �where the latter is used to calculateside loads�, then even in cases where acoustic distur-bances penetrate well upstream of the separation line,for computational purposes, we can neglect the asso-ciated pressure variation.

Finally, considering pressure variations due toboundary layer turbulence, the ratio of turbulent pres-sure variations to the mean pressure is on the order of

p� / Pi=O�u�2 / U2�, where the latter, representing theratio of characteristic upstream random to mean ve-locities, is small.

On the downstream side of the instantaneousseparation line, a large, subsonic, near-wall separationzone exists.1,4,36 Acoustic pressure fluctuations at andnear the nozzle exit, as well as those within the sepa-ration zone, can thus propagate upstream, and indeed,these fluctuations are implicated as the primary sourceof the low frequency, large amplitude separation linemotions noted above. �High frequency, small ampli-tude jitter is also observed39 and appears to be pro-duced by advection of vorticity through the foot of theseparation-inducing shock.� Since spatial pressurevariations upstream of the instantaneous separationline appear to be small �as argued above�, we can usethe characteristic magnitude of the random down-stream pressure fluctuations, �p�, as a useful estimateof the maximum pressure fluctuations extant withinthe entire shock-boundary layer interaction zone.

An estimate for �p� follows via two equivalentroutes: �i� focus on the axial dynamics of boundarylayer particles immediately downstream of the instan-



taneous separation line and scale particle inertiaagainst the net axial pressure force29 or �ii� scale axialinertia, �ut��2Ls /�s

2, against pressure, Px��p� /Ls,in the Navier–Stokes equations. Either approach leadsto

�p�/P2 � �p�/Pa � ��aLs/�s2�/Pa � 1, �38�

where the downstream density and pressure, �2 andP2, are approximately equal to �altitude-dependent�ambient values, �a and Pa. Thus, over the boundarylayer-shock interaction zone, spatial pressure varia-tions on either side of the instantaneous separationline are small relative to the instantaneous �altitudedependent� downstream mean, P2�H�t�� and again,are thus small relative to P2− Pi�.

�v� Returning to the model, we express the probability ofobserving any given instantaneous separation lineshape, s� , t�, as a joint probability density

ps = ps�s1,s2, . . . ,sN� , �39�

over the N-dimensional set of random variables de-scribing the shape

�s1 = s� 1,t�, s2 = s� 2,t�, s3 = s� 3,t�, . . . , sN

= s� N,t��

and assume that each member of the set�s1 ,s2 ,s3 , . . . ,sN� is

�a�independent,�b�Gaussian, and�c�has the same �altitude-dependent� variance, var�si�

=�s2=�s

2�H�.

Considering first assumption �a�, we expect thatunder conditions where large downstream azimuthalacoustic modes are not excited �such as those impli-cated in tee-pee separation patterns,1,5 for example�,this assumption is approximately valid. In addition,this assumption leads to considerable mathematicalsimplification. Assumption �b� is consistent with ear-lier observations,4,43 while �c� appears reasonable,again under conditions where nozzle shape and com-bustion are nominally symmetric, and where down-stream flow asymmetries are small. Taken together,and as shown below, these assumptions lead to theo-retical results, given in Eq. �46� below, that are con-sistent with observed side-load statistical properties.

�vi� Finally, when moving from the discrete to continuouslimit, � →d , and consistent with assumptions �va�and �vc� above, we assume that the instantaneousseparation line, s� , t�, is delta correlated in

�s� ,t�,s� �,t��s = �s2�� − �� , �40�

where � · �s denotes an ensemble average over thespace of all separation line shapes.

D. Side-load statistical properties

Having proposed a statistical model of separation linedynamics, we can now calculate side-load statistical proper-ties, specifically ensemble averages of the lateral side-loadcomponents, �Fsy�s and �Fsz�s, and importantly for presentpurposes, the mean squared side-load amplitude, �A2�s.

Based on assumptions �va�—�vc� above, the joint prob-ability density, ps, associated with the instantaneous randomseparation line shape is given by

ps�s1,s2, . . . ,sN� = �I

pI =1

�2�s�N/2

�exp�−s1

2 + s22 + . . . + sN

2

2�s2 � . �41�

Expressing ps as the product of the N Gaussian pdfs, pI, I=1,2 , . . . ,N, where each pI, given by

pI�sI� =1

�2�s2exp�−

sI2

2�s2� , �42�

is again associated with the independent random line dis-placement, sI=s� I , t�.

Ensemble averages of the instantaneous side-load com-ponents Fsy and Fsz then follow as:

�Fsy�H�t��s = R�H�t��Pi�H�t��

− Pa�H�t��0

2

sin �s� ,t��sd �43�

and

�Fsz�H�t��s = R�H�t��Pi�H�t��

− Pa�H�t��0

2

cos �s� ,t��sd , �44�

where we approximate the downstream pressure, P2�H�t�� asthe instantaneous ambient pressure, Pa�H�t��.29,36 In order toevaluate these averages, express the kth realization of, e.g.,Fsy

�k�, in discrete form as

Fsy�k��s1,s2, . . . ,sN� = R�H�t��Pi�H�t�� − Pa�H�t��

��I=1

N

s�k�� I�sin� I�� , �45�

where s�k�� I� is the associated separation line displacementat I. Taking the ensemble average term by term, and notingthat

�sI�s = �−�

� ��

�

¯�−�

�

sIps�s1,s2, . . . ,sN�ds1ds2 . . . dsN

= 0,

then leads to the result that ensemble averaged values of bothside-load components are zero



�Fsy�H�t��s = 0, �Fsz�H�t��s = 0. �46�

The altitude-dependent average squared side-load amplitude,�A2�H�t��s, is finally determined as follows:

A2�H�t�� = Fsy2 �H� + Fsz

2 �H� = R2�H�t��Pi�H�t��

− Pa�H�t��2��0

2

s� �cos d �2

+ �0

2

s� �sin d �2�or

A2�H�t�� = R2�H�t��Pi�H�t�� − Pa�H�t��2

��0

2 �0

2

s� �s� ��cos� − ��d d �� . �47�

Taking the average �A2�s, and using Eq. �40�, we get

�A2� = �A2�H�t�� = 2�s2R2�H�t��Pi�H�t�� − Pa�H�t��2.

�48�

Taking �s as the experimentally observed �nominal� lengthof the shock interaction zone, Ls, the sideload model isclosed finally by using the right side of Eq. �48� in Eq. �36�,yielding the parameter � in Eq. �31� ��or Eq. �33��.

IV. RESULTS AND DISCUSSION

The rocket dynamics model used for numerical experi-ments corresponds to the translational and rotational equa-tions of motion, Eqs. �12�–�14� and �23�–�25�. During theside-loading period, 0� t�T, this nonlinear coupled systemis forced by random nozzle side loads. In order to simulateany given side-load history, a three-step Monte Carlo ap-proach is employed. First, at any instant t, the instantaneousmean separation line location, xs�t�, is determined using anapproach outlined in Keanini and Browm.29 Thus, an ap-proximate nozzle pressure ratio is first calculated as

NPR�t� =Po

Pa�H�t��=

Po

Pi

Pi

P2

P2

Pa�H�t��

= g�Mi�f�Mi,��c�t� , �49�

where g�Mi�= P0 / Pi, f�Mi ,��= Pi / P2, c�t�= P2 / Pa�H�t��, Pi

and P2 are pressures upstream and downstream of theseparation-inducing shock, � is the shock angle, and Mi isthe associated upstream Mach number. The implicit functiong�Mi� is obtained via the generalized quasi-one-dimensionalmodel of isentropic flow44 while f�Mi ,�� corresponds to thepressure ratio function for oblique shocks based on Keaniniand Brown separation criteria.29 The function c�t� capturesthe small down-stream pressure rise that drives flow withinthe near-nozzle-wall recirculation zone; generalizing, e.g.,the results from Keanini and Brown29 to the present case oftime-varying ambient pressure, we assume that c�t�=0.85.Similarly, and consistent with a number of shock angle mea-surements �see literature overview in Ref. 29�, we take theshock/flow deflection angle ��15.2°. Again, using the well-known shock relationship �see Eq. �50��, the turning/

deflection angle � can be easily expressed �especially for theinviscid flow outside the separating boundary layer� as afunction of incipient Mach number Mi and the shock angle ��Ref. 44�

tan � =2 cot ��Mi

2 sin2 � − 1�Mi

2�� + cos 2�� + 2. �50�

Thus, given NPR�t� and turning angle �, Eqs. �49� and �50�allow determination of the associated incipient upstreamMach number, Mi=Mi�t�. Given Mi�t�, the correspondingnozzle radius, R�t�, is determined using the area-Mach num-ber relation for quasi-one-dimensional isentropic flow. GivenR�t�, xs�t�, then follows from the known nozzle geometry.

Second, given the instantaneous mean separation lineposition, a single realization of the instantaneous separationline shape, s�� , t�, is generated incrementally: at any givenangular position � j = j�2�N−1, j=1,2 , . . . ,N, a separationline displacement, �sj =s�� j , t�, is determined by samplingthe cumulative distribution function associated with the dis-placement amplitude density, Eq. �42�.

Third, once N independent displacements,��s1�t� ,s2�t� , . . . ,�sN�t��, are thus computed, associated in-stantaneous side-load components, Fsy�H�t�� and Fsz�H�t��,are calculated via single realization �nonaveraged� versionsof Eqs. �43� and �44�. �Note: the instantaneous rocket alti-tude, H�t�, is determined via the vertical momentum Eq.�12�; this in turn allows determination of the ambient pres-sure, Pa�H�t��. In addition, the pressure Pi�t� is obtained us-ing the quasi-one-dimensional isentropic relation for P0 / Pi.�

Model simulations were performed using MATLAB/

SIMULINK. Single realization time histories of side loadingand associated translational and rotational displacementswere obtained by numerically integrating the governingequations using a fourth-order Runge Kutta algorithm.Model parameters, given in Table I, are representative ofthose associated with sounding rockets, e.g., thePeregrine45–47 and Black Brant.48 This choice was guided byvarious scaling arguments, all of which showed that side-load effects on rocket dynamics become increasingly promi-

TABLE I. Model simulation parameters and values.

Parameters Values

Mass of rocket, M �at t=0� 1200 kgNozzle exit radius, Re 0.25 mNozzle throat radius 0.05 mRadius of main rocket body, Ri 0.2 mNozzle divergent section angle 15°Length of rocket, L 10 mLocation of center of mass of rocket from the apex, b 5 mMoment of inertia about roll axis, Ixx �at t=0� 135 kg m2

Moment of inertia about yaw/pitch axis, I �at t=0� 10 000 kg m2

Combustion chamber pressure �stagnation pressure�, P0 7 MPaCombustion chamber temperature �stagnationtemperature�, T0 3600 KPolytropic exponent of exhaust gas 1.34Gas constant for exhaust gas 355.47 J /kg m2

Polytropic exponent of ambient air 1.4Gas constant for ambient air 287 J /kg K



nent with decreasing rocket size. �The most straightforwardof these proceeds as follows: form the ratio of characteristicside-load magnitude to characteristic thrust

Fs

�eue2Ae

��Pa − Pi�2Re�s

kPeMe2Re

2 ��s

kMe2Re

,

where subscripts refer to values at the nozzle exit and whereRe is the nozzle exit radius. Here, we used Pa� Pt �where Pi

is the characteristic nozzle pressure immediately upstream ofthe separation-inducing shock�, as well as Pe� Pa �which isapproximately true while the separation-inducing shock lieswithin the nozzle�. Since Me

2 and �s are, in a order of mag-nitude sense, relatively fixed for a range of rocket nozzlesizes, then it is clear that relative side-load magnitudes in-crease with decreasing rocket size.�

As soon as rocket ascends from sea-level, an obliqueshock system is generated within its exhaust nozzle due tooverexpanded flow condition �the isentropic nozzle flow exitpressure being lower than the ambient atmospheric pressureat sea-level, i.e., with NPR �=Po / pa� being approximately70�. Due to adverse pressure gradient and nozzle geometry,flow separation from nozzle walls also occurs just about thesame location as the shock. With increasing altitude �i.e.,with increasing NPR�, the separation line and thus the shockcontinue to move downstream of the nozzle �as illustrated inFig. 6�. It is to be noted Fig. 6�a� only represents approxi-mate geometry of shock within the nozzle, where the radial�i.e., R�H�t�� and the axial positions �i.e., xs�H�t�� are re-lated to each other through nozzle geometry, thereby not cap-turing or emphasizing the detailed structure of complexshock-boundary layer interactions �refer to Ref. 1 for de-tails�. The flow within the nozzle continues to be overex-panded till an altitude of approximately 3.85 km. Once theshock exits the nozzle �i.e., at an approximate altitude of3.85 km�, the flow within the nozzle is fully isentropic—anyfurther discrepancy between the nozzle exit pressure and the

ambient pressure is compensated through a system of ob-lique shock diamonds or expansion fans past the nozzle exitplane. Since the nozzle flow is supersonic and fully isentro-pic past �4 km altitude, the pressure information past exitplane does not propagate upstream to the nozzle, and hencehas no influence on the in-nozzle side loads. So, even thoughFig. 7 illustrates that fully isentropic nozzle exit pressure�after �4 km altitude� is lower than the ambient pressureand continues to be so till an altitude of approximately 30km, pressure variations due to oblique compression shockdiamonds in this zone of the flight �i.e., from �4 to 30 km�do not affect the in-nozzle stochastic side-load generationprocess. Thus, side loads are generated within the nozzle aslong as the separation line and shock are confined to its in-terior.

Figure 5 shows a representative time history of both ran-dom side-load components, Fsy�t� and Fsz�t�. Several featurescan be noted. First, it is found that side-load magnitudes areonly one to two orders of magnitude smaller than the char-acteristic thrust; �Fsy ,Fsz�=O�103 N�, while thrust computedis of the order of 104–105 N. Thus, as anticipated �and aswill be shown�, side loads play a significant role in rocketdynamics. Second, side loading takes place only at low alti-tudes where ambient pressure remains high enough to forceoutside air into the nozzle. �The length of the side-load pe-riod, T, can be ascertained, e.g., from Fig. 11�b�, where it isseen that random forcing of the yaw rate, �y, ceases at ap-proximately 11 s.� Third, during the side loading period, aslight decay in side-load magnitudes is apparent. This can beexplained by referring to nonaveraged versions of Eqs. �43�and �44� along with Figs. 6 and 7: over 0� t�T, the pressuredifference, Pi�H�t��− Pa�H�t��, determining the side-loadmagnitude drops by roughly 39% while the nozzle radius,R�H�t��, increases by only 18%.

A representative set of single realization results, showingtime histories for: �i� position of the rocket’s center of mass,

FIG. 5. �Color online� In-nozzle stochastic side loads�n� vs rocket vertical altitude �km�.



�ii� center of mass translational velocity, and �iii� pitch, yawand roll rates are presented, respectively, in Figs. 8, 9, and11. The sets of results shown correspond to a single numeri-cal experiment and in all cases, for purposes of comparison,fully deterministic time histories obtained with random sideloads turned off are also included. The initial conditions usedin this and all other numerical experiments are as follows: allinitial translational and rotational velocities and displace-ments are zero. Several important observations can be made.

�i� As shown in Fig. 8, random side loads are capable ofproducing significant �random� lateral displacements,on the order of 1 km over the 25 s simulation period.As also shown, and as expected, no lateral displace-ment occurs when side loads are suppressed. Themagnitudes of observed displacements, in this and all

experiments, are of the appropriate scale, i.e.,

Fs�⁄ ��eve2Ae��MR� / �MRXo�→��Tf� /Xo�Tf�

�Fs� / ��eve2Ae�=O�10−2–10−3�. Here again, ��t�

represents either Yo�t� or Zo�t� and Tf is the total flightsimulation time, i.e., 25 s.

�ii� Due to the same scaling, the effect of side loads onvertical displacements, Xo�t�, and thus total displace-ment, ro�t�=�Xo

2+Yo2+Zo

2, is negligible; see Figs. 8�a�and 10.

�iii� The same simple scaling argument can be used tointerpret observed rocket velocity histories in Figs.

9�b� and 9�c�; here, ��Tf� / Xo�Tf�Fs� / ��eve2Ae�. Like-

wise, the scale of the random variation in the vertical

velocity component, Xo�t�, �relative to the no-side-

FIG. 6. �Color online� Nozzle shock location �a� radialposition of shock �Rshock in centimeter� vs its axial po-sition �Xshock in centimeter� in the nozzle �note this isonly an approximation of the shock structure in a realnozzle and is essentially based on or computed fromnozzle geometry�, �b� axial position �Xshock in centime-ter� of shock in nozzle vs rocket vertical altitude fromground �in km�, and �c� radial position of shock �Rshock

in centimeter� vs rocket vertical altitude from ground�in kilometer�.

FIG. 7. �Color online� Pressure �in kilopascal� vs alti-tude above sea-level �in kilometer�: pa is the ambientatmospheric pressure, pe is the nozzle exit pressurepostshock, and pe,up,shock is the pressure upstream of theshock.



load history� is on the same order �Fig. 9�a��. Closerinspection of Fig. 9�a�, however, yields that the verti-cal ascent velocity of the rocket is slightly lower whennozzle side loading is taken into account. This couldbe attributed to transfer of momentum from the lon-gitudinal direction �i.e., x-direction� to lateral direc-tions �i.e., y- and z-directions� as rocket undergoesyaw and pitch under the influence of these stochasticin-nozzle side loads. Qualitatively, the thrust-timecurves �equivalently vX,o versus time curves� for simi-lar small, single-stage rockets have been reported toexhibit similar characteristics as shown in Fig. 9�a�.47

�iv� As expected, and as shown in Figs. 9�b� and 9�c�,when side loads are turned off, lateral velocities re-main zero throughout any given simulation.

�v� The effect of side loading on total velocity and dis-placement is small and on the order of Fs� / ��eve

2Ae�;refer to Fig. 10. This simply reflects the dominance ofthe vertical velocity component relative to the lateralcomponents.

�vi� During the side loading period, pitch and yaw ratesexhibit random responses to the random internaltorques excited by side loads �refer to Fig. 11�; incontrast, and due to the lack of coupling between rolland side loading, the roll rate remains zero throughoutthe simulated flight given zero initial conditions onroll, pitch and yaw. Following the side-load period,the pitch and yaw rate evolution become wholly de-terministic, subject to a random initial condition at t=T�11 second �end of side loading period�. Under

FIG. 8. �Color online� Position time history of the rock-et’s center of mass O, as measured from an inertialXYZ reference frame �i.e., from ground�. �a� Time his-tory of X-position �i.e., vertical height or altitude fromground� of center of mass, �b� time history of Y-positionof center of mass, �c� time history of Z-position of cen-ter of mass, and �d� time history of radial position �ro�of center of mass from the origin of inertial XYZ frame,i.e., ro=�xo

2+yo2+zo

2.

FIG. 9. �Color online� Velocity time history of the rock-et’s center of mass O, as measured from an inertialXYZ reference frame �i.e., from ground�. �a� Time his-tory of velocity of center of mass in X-direction, �b�time history of velocity of center of mass in Y-direction,and �c� time history of velocity of center of mass inZ-direction.



the no-side-load scenario, since no internal or externaltorques are present, the rocket rotation remains essen-tially zero. It is interesting to note from both Figs. 11and 13 that once the side-loading period is over, thepitch and yaw rotation rates of the rocket tend tomove toward the zero mean, thereby emphasizing anunderlying rich dynamics associated with a “mean-reverting” process. The detailed analyses capturingthe nature of this “mean-reverting” process will beemphasized in subsequent publications.

The differences in the dynamic response of rigid-bodyrocket model for side loading and no side load scenarios arethus quite clear from Figs. 8–11. Although the rocket’scenter-of-mass altitude �i.e., Xo in Fig. 8�a�� and verticallaunch velocity �i.e., vX,o in Fig. 9�a�� are not affected much

by in-nozzle side loads, it’s the rocket lateral motion �i.e.,Yo , Zo , vY,o , vZ,o in Figs. 8 and 9� that is significantly in-fluenced by these side loads. The rocket during its attitude orascent thus continues to exhibit deviations �though slight/minor—refer to Fig. 10� from the path that it would havetaken had there been no side loads in the nozzle. Also, it isinteresting to note that since the side loads in y- andz-directions exhibit nearly same characteristics �or trends�, asdepicted in Fig. 5, the y- and z-direction motions of therocket tend to be similar, as could be inferred more clearlyfrom Figs. 13 and 15, thereby implying the stochastic distri-bution of side loads do not induce any preference iny-direction rocket motion over z-direction motion or vice-versa.

It is clear that Figs. 8–11 only represent a stochastic

FIG. 10. �Color online� Influence of nozzle side loadson path and speed of rocket during its attitude. �a� per-centage change in the path of rocket’s center of mass,i.e., 100ro−ro,NO /ro,NO% where ro=�xo

2+yo2+zo

2 is theradial position of rocket’s center of mass measuredfrom the origin of the inertial XYZ system �b� percent-age change in the speed of rocket’s center of mass, i.e.,100vo− voNO / voNO%, where vo=�xo

2+ yo2+ zo

2 is thespeed and the subscript “NO” refers to the no side-loadscenario.

FIG. 11. �Color online� Time history of rocket angularvelocity, as measured in body-fixed xyz reference frameattached to center of mass O. �a� Time history of rollangular velocity about x-axis, �b� time history of yawangular velocity about y -axis, and �c� time history ofpitch angular velocity about z-axis.



sample for the attitude dynamics of rocket. In order to betterunderstand the stochastic effects of side-load generation pro-cess on rocket attitude dynamics, numerous �i.e., 100� simu-lations were performed to get a collection of stochasticsamples. Later, several standard tests were performed onthese realizations �especially those related to or accountingfor the lateral motion of rocket� to capture the underlyingstochastic distribution characteristics. Figure 12 shows a col-lection of time histories �stochastic realizations� of the yawangular velocity �y of rocket obtained from 100 simulations.Knowing this collection of data for yaw angular velocity, it iseasy to obtain the mean time history of �y as well as thedeviations from it �refer to Fig. 13�. It could be inferred fromFig. 13 that with increasing �or large� number of stochasticsamples, the mean time history of �y would correspond to

the case of no side loading scenario, however, the error mar-gins or bounds �dependent on the standard deviation, ��would still be significant. This implies that concluding theaverage effect of side loads on yaw motion of rocket to benull would be erroneous as deviations from the mean behav-ior are not insignificant. The variance varies with time—growing steadily as long as shock is within the nozzle andside loads are being generated, and later decaying as shockgoes past the exit of nozzle plane �as side loads no longerexist once the shock escapes the nozzle, thereby forcing therocket to enter a stabilizing state that it would have seen ifthe side loads were not present at all�. Also, Fig. 12 depictstime slices of the collection of �y �yaw angular velocity�samples at 1, 2.5, 7, 15, and 25 s. The sample data at thesetime slices were analyzed to capture the underlying stochas-

FIG. 12. �Color online� Time histories �i.e., stochasticrealizations� of yaw angular velocity, �y, of the rocket:complete collection of �y samples from 100 simulationruns along with the time slices at 1, 2.5, 7, 15, and 25 s.

FIG. 13. �Color online� Time history of mean �i.e., ��yaw angular velocity, �y, and pitch angular velocity, �z,of the rocket and the errors or deviations �i.e.,meanstandard deviation �i.e., �� based on collectionof �y and �z samples for 100 simulations.



tic distribution. Using normal probability plots �refer to Fig.14� and the Anderson–Darling test it is seen that the under-lying probability distributions of yaw angular velocity at ev-ery time slice is Gaussian in nature. This is a critical obser-vation as it implies the existence of a unique Chapman–Kolmogorov equation, which once identified/formulated,could be used to analyze the time-evolution of probabilitydistributions of rocket attitude dynamic indices �especiallythe lateral motion parameters arising from stochastic sideloads�. Similar observations could be inferred from Figs. 15and 16. Again, it is clearly evident from Figs. 13 and 15 thatside loads do not induce any preference in the y- andz-direction motion parameters, whether they are yaw and

pitch angular velocities of the rocket about its center of massor the lateral translational displacements and velocities of thecenter of mass.

V. CONCLUSIONS

A long standing, though previously unsolved problem inrocket dynamics, rocket response to random, altitude-dependent nozzle side loads, has been investigated. Numeri-cal experiments, focused on determining single-realizationand ensemble average translational and rotational rocket dy-namics, incorporate a distributed mass, six-degree of free-

FIG. 14. �Color online� Normal probability plots for thecollection data of yaw angular velocity, �y, and pitchangular velocity, �z, at time slices of 1, 2.5, 7, 15, and25 s.

FIG. 15. �Color online� Time history of mean �i.e., ��lateral motion parameters for the center of mass of therocket and the corresponding errors or deviations �i.e.,meanstandard deviation �i.e., �� based on the collec-tion of stochastic samples for 100 simulations. �a� Meanpath/trajectory for center of mass in the Y-directionalong with the deviations, �b� mean path/trajectory forcenter of mass in the Z-direction along with the devia-tions, �c� mean velocity of center of mass in theY-direction along with the deviations, and �d� mean ve-locity of center of mass in the Z-direction along withthe deviations.



dom rocket model, representative of small, sounding-rocket-scale rockets. The principal contributions and findings are asfollows.

�1� A relatively simple, physically consistent model of ran-dom separation-line motion within rigid �nonvibrating�nozzles is developed. By circumventing difficult experi-mental �or numerical� determination of space-dependentin-nozzle pressure correlations, the proposed model of-fers distinct advantages over Dumnov’s34 widely usedapproach. Specifically, in exploiting well-known, nomi-nally universal statistical properties associated with therandom motion of shock-separated boundary layers, themodel allows analytical determination of altitude-dependent side-load statistics and straightforward MonteCarlo simulation of individual side-load histories.

�2� Scaling indicates that as rocket size decreases, side loadsplay an increasingly prominent role in rocket dynamics.For example, numerical experiments show that duringshort �25 s� simulated flight periods, the model rocketcan experience random, side-load-driven transverse dis-placements on the order of several kilometers. Likewise,side loads are found capable of inducing significant ran-dom pitch and yaw rates and displacements.

�3� During the low-altitude side-load period �approximately3.85 km�, pitch and yaw rates exhibit rapid increases instochasticity, as indicated by observed variances; similarbehavior is observed for lateral velocities. Followingnozzle expulsion of the side-load-inducing shock, how-ever, side-loads cease; nevertheless, subsequent lateraltranslational dynamics, as well as pitch and yaw rota-tional dynamics, remain subject to the stochasticity gen-erated during the side-load period. In the case of post-side-load pitch and yaw rate variances, these exhibit aslow decay toward zero. Conversely, lateral translational

velocity variances grow at an approximate quadratic ratewith altitude.

The implication of the results from this rocket modelsimulation is clearly twofold: first, rocket attitude-dynamicsmodels not incorporating side loads will predict erroneouslaunch trajectory and rigid-body rocket motion, which con-sequently degrades the controller performance owing togreater �often redundant� thrust and attitude control effortand lack of compensation for the stochastic loading onnozzle walls. Subsequent work will include discussion ontheoretical stochastic mechanics of a rocket under randomside loads, controller design to compensate for undesirableeffects of side loads, and development of more enhanced/detailed rocket dynamics model.

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FIG. 16. �Color online� Normal probability plot for thecollection data of lateral motion parameters of the rock-et’s center of mass at time slices of 1, 2.5, 7, 15, and 25s. �a� Normal probability plot for yo, �b� normal prob-ability plot for zo, �c� normal probability plot for vy,o

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Inﬂuence of nozzle random side loads on launch vehicle ... · Inﬂuence of nozzle random side...

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